Friday, January 27, 2023
01/27/2023 - 9:30am
A relaxed-pace seminar on impromptu subjects related to the interests of the audience.
Everyone is welcome.
The subjects are geometry, probability, combinatorics, dynamics, and more!
01/27/2023 - 12:00pm
It is an open question whether mapping class groups of a closed orientable surface is linear. One natural representation of a mapping class group into a linear group is its symplectic representation, which is induced from the action of the mapping class group on the first homology of the surface. It is well-known that the symplectic representation is surjective onto the integral lattice.
It is a natural question that what we can observe when we restrict the symplectic representation on the set of pseudo-Anosov mapping classes. Thurston proved that there are plenty of pseudo-Anosovs in the kernel of the symplectic representation, employing his construction of pseudo-Anosovs from filling multicurves.
In this talk, we show that the surjectivity still holds after restricting the symplectic representation on the set of pseudo-Anosovs. On the other hand, we also show that the surjectivity does not hold on the set of pseudo-Anosovs with orientable invariant measured foliations. This is the joint work with Hyungryul Baik and Inhyeok Choi, answering the question of Ursula Hamenstädt.
01/27/2023 - 2:00pm
An interesting feature of General Relativity is the presence of singularities which can happen in even the simplest examples such as the Schwarzschild spacetime. However, in this case the singularity is cloaked behind the event horizon of the black hole which has been conjectured to be generically the case. To analyze this so-called Cosmic Censorship Conjecture Penrose proposed in 1973 a test which involves Hawking’s area theorem, the final state conjecture and a geometric inequality on initial data sets (M,g,k). For k=0 this Penrose inequality has been proven by Huisken-Ilmanen and by Bray using different methods, but in general the question is wide open. Huisken-Ilmanen’s proof relies on the Hawking mass monotonicity formula under inverse mean curvature flow (IMCF), and the purpose of this talk is to generalize the Hawking mass monotonicity formula to initial data sets. For this purpose, we start with recalling spacetime harmonic functions and their applications which have been introduced together with Demetre Kazaras and Marcus Khuri in the context of the spacetime positive mass theorem.